Whenever we take a random sample from a population having variance σ, we know that the sample mean has variance given as σ/√n where n is the sample size. But this is only true if we assume that:
- The sample has been chosen with replacement.
- The population size is infinite.
Both of these assumptions are false in real-life situations. But the above estimate for the variance of the sampling distribution still works as an approximation as long as the sample size is less than 5% of the total population.
If the sample size exceeds 5% of the population then we need to multiply the standard deviation of the sample mean by a correction factor called the FPC (Finite population correction factor).
Formula for the finite population correction factor:
The FPC can be calculated using the formula,
Thus the formula for standard error of sampling mean becomes,
And the formula for standard error of sampling proportion becomes,
Example 1: Suppose that a population of size 10000 has a variance of 100. We want to estimate the average height of people in the population. If a sample of size 2500 is selected from the population then calculate the standard error of the sample mean.
Solution: The formula for standard error is given as S.E = σ/√n but since the sample is of size greater than 5% of the population (=500) therefore we must apply the correction factor to the standard error.
The formula becomes, S.E = √[(N-n)/(N-1)]* σ /√n = √(7500/9999) * 10/50 = 0.17321
Example 2: Suppose that we have a population of size 1000 that has a variance of 25. We obtain an estimate for the proportion ‘p’ of blue-eyed people in the population as p=0.5. If we have drawn a sample of size 100 from the population then calculate the standard error of the population proportion
Solution: The formula for standard error is given as, S.E = √p(1-p)/n but since the sample is of size greater than 5% of the population (=50) therefore we must apply the correction factor to the standard error.
The formula becomes, S.E = √[(N-n)/(N-1)]* √p(1-p)/n = √(900/999) * √(0.5*0.5/100) = 0.0474
